Abstract
Applying a method of the cluster expansion developed in a study of statistical mechanics, a new mathematical model has been structured to account for the reason that a time series of transaction signs in financial markets has a long memory property. A basic assumption for the model was that investors split their hidden orders into small pieces before execution. The effect of public information also was taken into consideration. A mathematical expression of investors’ investment behavior generates a discrete time stochastic process of cumulative transaction signs. The strong law of large numbers holds for the process: it converges to a trend term almost surely. The distribution of the fluctuation around the trend term weakly converges to the distribution of a superposition of a stochastic integral with respect to a Brownian motion and stochastic integrals with respect to a fractional Brownian motions with Hurst exponents greater than one-half. Namely, increments of the derived process have a long memory property.
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Acknowledgement
The authors thank the Yukawa Institute for Theoretical Physics at Kyoto University. Discussions during the YITP workshop YITP-W-15-15 on ``Econophysics 2015'' were useful to complete this work.
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J. Murai is partially supported by Grant-in-Aid for Scientific Research (C) No. 15K01190.
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Murai, J. A model of transaction signs with order splitting and public information. Evolut Inst Econ Rev 13, 469–480 (2016). https://doi.org/10.1007/s40844-016-0050-5
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DOI: https://doi.org/10.1007/s40844-016-0050-5